The application of dynamic programming to slope stability analysis

Transcription

1 830 The application of dynamic programming to slope stability analysis Ha T.V. Pham and Delwyn G. Fredlund Abstract: The applicability of the dynamic programming method to two-dimensional slope stability analyses is studied. The critical slip surface is defined as the slip surface that yields the minimum value of an optimal function. The only assumption regarding the shape of the critical slip surface is that the surface is an assemblage of linear segments. Stresses acting along the critical slip surface are computed using a finite element stress analysis. Assumptions associated with limit equilibrium methods of slices related to the shape of the critical slip surface and the relationship between interslice forces are no longer required. A computer program named DYNPROG was developed based on the proposed analytical procedure, and numerous example problems have been analyzed. Results obtained when using DYNPROG were compared with those obtained when using several well-known limit equilibrium methods. The comparisons demonstrate that the dynamic programming method provides a superior solution when compared with conventional limit equilibrium methods. Analyses conducted also show that factors of safety computed when using the dynamic programming method are generally slightly lower than those computed using conventional limit equilibrium methods of slices; however, as Poisson s ratio approaches 0.5, the computed factors of safety from the dynamic programming method and the limit equilibrium method appear to become similar. Key words: dynamic programming, slope stability, stress analysis, optimization theory, limit equilibrium methods of slices. Résumé : On étudie l applicabilité de la méthode de programmation dynamique à des analyses de stabilité de talus à deux dimensions. La surface critique de glissement est définie comme la surface de glissement qui donne la valeur minimale d une fonction optimale. La seule hypothèse concernant la forme de la surface critique de glissement est que la surface est constituée d un assemblage de segments linéaires. Les contraintes agissant le long de la surface critique sont calculées au moyen d une analyse de contraintes par éléments finis. Des hypothèses reliées aux méthodes d équilibre limite des tranches par rapport à la forme de la surface critique de glissement de même que la relation entre les forces intertranches ne sont plus requises. On a développé un programme d ordinateur appelé DYNPROG basé sur la procédure analytique proposée et on a analysé de nombreux exemples de problèmes. Les résultats obtenus en utilisant DYNPROG ont été comparés avec ceux obtenus au moyen de plusieurs méthodes d équilibre limite bien connues. Les comparaisons démontrent que la méthode de programmation dynamique fournit une meilleure solution par rapport aux méthodes conventionnelles d équilibre limite. Les analyses réalisées montrent aussi que les coefficients de sécurité calculés au moyen de la méthode de programmation dynamique sont généralement légèrement plus faibles que les coefficients de sécurité calculés au moyen des méthodes conventionnelles d équilibre limite des tranches; cependant, lorsque le coefficient de Poisson s approche de 0,5, les coefficients de sécurité calculés au moyen de la méthode de programmation dynamique et ceux de la méthode d équilibre limite semblent devenir semblables. Mots clés : programmation dynamique, stabilité des talus, analyse des contraintes, théorie d optimisation, méthodes d équilibre limite des tranches. [Traduit par la Rédaction] Pham and Fredlund Introduction A conventional slope stability analysis involving limit equilibrium methods of slices consists of the calculation of Received 12 July Accepted 25 March Published on the NRC Research Press Web site at on 11 August H.T.V. Pham. Department of Civil and Construction Engineering, Iowa State University, Ames, IA , U.S.A. D.G. Fredlund. 1 Department of Civil Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9, Canada, 1 Corresponding author ( dgf681@young.usask.ca). the factor of safety for a specified slip surface of predetermined shape and the determination of the location of the critical slip surface with the lowest factor of safety. To render the inherently indeterminate analysis determinate, conventional limit equilibrium methods generally make use of assumptions regarding the relationship between the interslice forces. These assumptions become disadvantages to limit equilibrium methods, since the actual stresses acting along the slip surface are quite approximate and the location of the critical slip surface depends on the shape assumed by the analyst. The assumptions related to the interslice force function in limit equilibrium methods are unnecessary when a finite element stress analysis is used to obtain the normal and shear Can. Geotech. J. 40: (2003) doi: /T03-033

2 Pham and Fredlund 831 stresses acting at the base of slices (Fredlund and Scoular 1999). A stress analysis provides normal and shear stresses through the use of the finite element numerical method with a switch on of the gravity forces. Subsequently, the equation for the factor of safety becomes linear. Assumptions regarding the uncertainty of the shape of the critical slip surface can be omitted when an appropriate optimization technique is introduced into the analysis. Optimization techniques have been developed by several researchers for over two decades and have provided a variety of approaches to determine the shape and location of the critical slip surface (Celestino and Duncan 1981; Nguyen 1985; Chen and Shao 1988; Greco 1996). Each approach has its own advantages and shortcomings. The main shortcoming associated with these approaches, however, is that the actual stresses within a slope are quite approximate. This disregard for a more accurate assessment of the stresses can lead to inaccuracies in the computation of the factor of safety and an inability to analyze more complex problems. The dynamic programming method can be combined with a finite element stress analysis to provide a more complete solution for the analysis of slope stability because the technique overcomes the primarily difficulties associated with limit equilibrium methods. The disadvantage of the dynamic programming approach is that there are more variables to specify for the analysis, such as Poisson s ratio and the elastic moduli of the soils involved. The dynamic programming method for a slope stability analysis has not been widely used in engineering practice primarily because of the complexity of the formulation and the lack of verification of the computed results. Baker (1980) introduced an optimization procedure that utilized the algorithm of the dynamic programming method to determine the critical slip surface. In this approach, the associated factors of safety were calculated using the Spencer (1967) method of slices. Yamagami and Ueta (1988) enhanced Baker s approach by combining the dynamic programming method with a finite element stress analysis to more accurately calculate the factor of safety (Fig. 1). The critical slip surface was assumed to be a chain of linear segments connecting two state points located in two successive stages. The resisting and the actuating forces used to calculate an auxiliary function were determined from stresses interpolated from Gaussian points within the domain of the problem. Yamagami and Ueta analyzed two example problems to illustrate the proposed procedure. Zou et al. (1995) proposed an improved dynamic programming technique that used essentially the same method as that introduced by Yamagami and Ueta (1988). The modification made by Zou et al. was that the critical slip surface might contain a segment connecting two state points located in the same stage. The stability of a trial dam in Nong Ngu Hao, Bangkok, Thailand, was analyzed as part of the study of the proposed procedure. The objective of this research program is to study the use of the dynamic programming method in solving practical slope stability problems. The analytical procedure behind the dynamic programming method is mainly based on the research of Yamagami and Ueta (1988). A computer program named DYNPROG was developed to interface with a general partial differential equation solver known as FlexPDE (PDE Solutions Inc. 2001) to determine the stress states in the soil mass and then determine the shape and location of the critical slip surface and the corresponding factor of safety. Numerous example problems have been solved using DYNPROG. Examples studied include homogeneous slopes, layered slopes, and a case history. The results obtained from the analyses were compared with results from several wellknown limit equilibrium methods of slices (Fredlund and Krahn 1977). 2. Background Bellman (1957) introduced a mathematical method called the dynamic programming method. One of the objectives of the dynamic programming method was to maximize or minimize a function. The dynamic programming method has been widely used in various fields other than geotechnical engineering. Baker (1980) appears to be the first to apply the optimization technique in the analysis of the stability of slopes. 2.1 Definition of the factor of safety For an arbitrary slip surface AB, as shown in Fig. 2, the equation for the factor of safety can be defined as [1] F s = B f A B τ dl A τ dl where τ is the mobilized shear stress along the slip surface, τ f is the shear strength of the soil, and dl is an increment of length along the slip surface. It is assumed that the critical slip surface can be approximated by an assemblage of linear segments. Each linear segment connects two state points located in two successive stages. The stage state system forms a grid consisting of rectangular elements called the search grid. The rectangular elements formed by the search grid are called grid elements. In this discretized from, the overall factor of safety for the slip surface AB is defined as follows: [2] F n i s = = 1 n i= 1 τ L f i τ L i i i where n is the number of discrete segments, τ i is the shear stress actuated, τ fi is the shear strength, and L i is the length of the segment. 2.2 Theory of the dynamic programming method A minimization is necessary for the value of the factor of safety, F s, in eq. [2]. It was shown by Baker (1980) that the minimum of F s in eq. [2] can be found by using an auxiliary function G. The auxiliary function is also known as the return function, and it can be defined as follows (Fig. 3): [3] G = ( Ri FS s i ) n i= 1

3 832 Can. Geotech. J. Vol. 40, 2003 Fig. 1. Search for the critical slip surface based on the dynamic programming method (after Yamagami and Ueta 1988). Fig. 2. An arbitrary surface AB in a discretized form. where S i are actuating forces acting on the ith segment of the slip surface, R i are resisting forces acting on the ith segment of the slip surface, and n is the total number of discrete segments making up the slip surface. The minimum value of the auxiliary function is G m and is defined as [4] Gm = min ( Ri FS s i) n i= 1 Along the ith segment, the shear strength for a saturated unsaturated soil can be calculated using the following equation (Fredlund and Rahardjo 1993): b [5] τf = c + ( σn ua)tan φ + ( ua uw)tan φ i where c, φ, and φ b are the shear strength parameters of a saturated unsaturated soil; (σ n u a ) is the net normal stress acting on the ith segment; and (u a u w ) is the matric suction. The normal and shear stresses acting on the ith segment can be computed from a stress analysis as follows: [6] σ n = σ x sin 2 θ + σ y cos 2 θ τ xy sin 2θ 2 2 ( [7] τ τ θ θ) σ y σ x) n = xy(sin cos sin 2θ 2 where σ n and τ n are the normal and shear stresses acting on the ith segment, respectively; θ is the inclined angle of the ith segment with the horizontal direction; and σ x, σ y, and τ xy are the normal and shear stresses acting in the x- and y-coordinate directions. These stresses can be determined using a finite element stress analysis that uses any particular soil behaviour model. If the density of the search grid is sufficiently fine, it can be assumed that stresses are constant within a small grid element. These constant stresses are signified by stresses at the centre points of the grid element. Consequently, the resisting and actuating forces acting on the ith segment of a slip surface can be calculated as follows (Fig. 3): [8] R = R = τ l i ne ne ij f ij = 1 ij = 1 ne ij ij = [ c + ( σ u )tan φ + ( u u )tan φ ] l ne ij = 1 [9] S = S = τ l i ij ij ij ij = 1 ij = 1 ij ne ij n a ij a w ij b ij

4 Pham and Fredlund 833 Fig. 3. Actuating and resisting forces acting on the ith segment. where (ij) is a grid element travelled by the ith segment; τf and τ ij ij and τ ij are the shear strength and shear stress actuated at the centre point of (ij), respectively; c ij, φ b ij, and φ ij are the strength parameters of the saturated unsaturated soil within (ij); ne is the number of (ij); and l ij is the length of the ith segment limited by the boundary of (ij). An optimal function, H i (j), obtained at state point {j} located in stage [i] is introduced. The optimal function, H i (j), is defined as the minimum of the return function, G, calculated from a state point for the initial stage to state point {j} located in stage [i]. According to the principle of optimality (Bellman 1957), the optimal function, H i+1 (k), obtained at state point {k} located in stage [i +1] is defined as [10] Hi= 1( k) = Hi() j + Gi(, j k) where Gi( j, k)is the return function calculated from state point {j} of stage [i] to state point {k} of stage [i +1]. At the initial stage, the value of the optimal function, H 1 (j), is equal to zero. That is, [11] H ( j) = 0 j = 1 NP 1 1 where NP 1 is the number of state points in the initial stage. At the final stage (i.e., i = n + 1), the optimal function, H n+1 (k), must be equal to the minimum value of the return function, G m, that is [12] H ( j) = G = min ( R FS) j = 1 n+ 1 n m i s i NPn+ 1 i= 1 where NP n+1 is the number of state points located in the final stage. The optimal point in the final stage is defined as the state point at which the calculated optimal function is a minimum. From the optimal state point {k} found in the final stage, the optimal state point {j} located in the previous stage is also determined. The optimal path defined by connecting optimal state points located in every stage is eventually found by tracing back from the final stage to the initial stage. This optimal path defines the critical slip surface. The value of the overall factor of safety, F s, in eq. [3] has not been defined in advance and therefore an initial value must be assumed. The trial value of F s is updated using the value of F s evaluated after each trial of the search. The optimization process will stop when a predefined convergence is reached. 2.3 Finite element stress analysis using FlexPDE The general partial differential equation solver known as FlexPDE is a flexible computer program that can be used to solve single or coupled sets of partial differential equations. FlexPDE allows the user to pose a problem in a compact problem-oriented form and proceed directly to a graphical presentation of the solution, without digressing to program the finite element method. For the plane strain condition (i.e., strain in the z-coordinate direction ε z = 0), a soil element subjected to its body forces has partial differential equations representing the stress balance defined as follows: σ τ [13] x xy + + Fx = 0 x y [14] τ x xy + σ y y + F = 0 y where σ x and σ y are normal stresses in the Cartesian x- and y-coordinate directions, respectively; τ xy is the shear stress in the xy plane; and F x and F y are body forces in the x- and y- coordinate directions, respectively. Partial differential eqs. [13] and [14] can be solved using FlexPDE along with specified boundary conditions. The domain of the problem is automatically divided by the computer program, FlexPDE, into triangular elements. The

5 834 Can. Geotech. J. Vol. 40, 2003 Fig. 4. The analytical scheme of the dynamic programming method in slope stability analyses. variables are presented by a simple polynomial equation over the problem domain. FlexPDE uses a Galerkin finite element model, with quadratic- or cubic-based functions involving nodal values of system variables. The stresses are evaluated and stored at Gaussian points over the domain of the problem when solving eqs. [13] and [14] for a specific problem. FlexPDE can interpolate and export stresses from Gaussian nodes to the nodes of any arbitrary grid defined by the user. The stresses at the centre point of each grid element are interpolated using the interpolation shape functions. The stress-interpolation process is done prior to the performance of the dynamic programming search. 2.4 Description of the computer program DYNPROG The analytical scheme of the dynamic programming method for performing a slope stability analysis is illustrated in Fig. 4. The computer program DYNPROG was developed to solve a slope stability problem using the following steps. (1) Input the geometry data and soil properties of the problem. (2) Import the output grid with corresponding nodal stresses from FlexPDE. (3) Define a search boundary using the output grid imported from FlexPDE as the search grid. (4) Interpolate stresses at the centre point of each grid element from nodal stresses. (5) Assume an initial factor of safety, F s (i.e., F s = 1). (6) Launch the search from all state points located in the initial stage. (7) Generate the first trial segment of the slip surface by connecting all state points of the initial stage to all state points located in the second stage. (8) Calculate the values of the optimal function obtained at all state points of the second stage using eqs. [10] and [11] and the assumed factor of safety, F s. The number of optimal functions to be calculated at one state point of the second stage is equal to the number of state points located in the initial stage. (9) Determine the minimum value of the optimal function at each state point in the second stage. The corresponding state point in the previous stage (i.e., the initial stage for the first segment) is identified. (10) Proceed to the next stage with the same routine until the final stage is reached. (11) Compare the values of the optimal functions obtained at all state points of the final stage and determine the state point at which the corresponding value of the optimal function is a minimum. The determined state point will be the first optimal point of the optimal path. (12) Trace back to the previous stage to find the corresponding state point with the first optimal point. This corresponding state point will be the second optimal point of the optimal path. (13) Keep tracing back to the initial stage to determine the entire optimal path. (14) Evaluate the actual factor of safety corresponding to the optimal path obtained from step 13 using eq. [2]. A new value for the factor of safety is calculated based on the initially assumed and the actual factors of safety. (15) Repeat the procedure until the difference between the assumed and the actual factor of safety is within the convergence criterion, δ, defined prior to the performance of the optimization process. (16) Define the actual critical slip surface by determining the entry and exit points of the critical slip surface. These points are found at the intersections of the optimal path with the physical boundary of the slope. 2.5 Restriction applied to the shape of the critical slip surface The shape of the critical slip surface must be kinematically admissible. Baker (1980) assumed that the critical slip surface must be concave. Therefore, the condition applied to the shape of the critical slip surface proposed by Baker was that the first derivative calculated from the crest to the toe of the curve that represents the critical slip surface must be greater than or at least equal to zero. Kinematic restriction conditions were not mentioned in Yamagami and Ueta (1988). Zou et al. (1995) stated that a check must be made to assure that the critical slip surface is kinematically admissible. There was no further comment regarding how this

6 Pham and Fredlund 835 Fig. 5. Kinematical restrictions applied to the shape of the critical slip surface. check should be applied, however. The authors of this article suggest that kinematical restrictions play an important role in the applicability of the dynamic programming method in slope stability analysis. Using appropriate kinematical restrictions prevents the shape of the critical slip surface from being unreasonable. Theoretically, when failure takes place the resisting force and the actuating force along the slip surface must be in contrary directions. The resisting force must always act in the direction opposite to the mass movement. At the same time, the actuating force must be in the same direction as the movement (Fig. 5). The kinematical restriction applied to the shape of the critical slip surface in this study is that if the actuating force calculated is in a contrary direction to the anticipated direction of mass movement, then the entire trial segment in which the actuating force is being calculated will be eliminated from the search. In other words, a trial segment will be eliminated from the optimization search if the actuating and resisting forces are found having the same sign. Applying this condition to the optimization procedure will eliminate all trial segments that constitute kinky-shaped slip surfaces. 3. Application of the dynamic programming technique A number of example problems were studied to illustrate the flexibility of the dynamic programming technique and to compare the computed factors of safety with those obtained from other methods. 3.1 Simple homogeneous slope The stability of a simple homogeneous slope at 2:1 is examined in this section. The stresses and pore-water pressures were computed using FlexPDE. The use of FlexPDE for solving saturated unsaturated seepage problems was performed by Nguyen (1999). The stress strain relationship was assumed to be linear elastic. Two conditions were considered related to the distribution of the pore-water pressures in the slope. In the first case, the slope was referred to as being in the wet condition with the groundwater table passing through the toe of the slope. The second case was referred to as the submerged condition where the slope was partly submerged in water. In the submerged condition, the water table was 2 m deep at the toe of the slope and the groundwater table was 4 m below the crest surface (later shown in Fig. 12). Each case study solved in this section utilized two values of Poisson s ratio, namely µ = 0.33 and These values were considered as being a reasonable value and a limiting value, respectively, for soils. The unit weight of the soil was 18 kn/m 3. There were 12 sets of shear strength parameters used for each value of Poisson s ratio. With the combination of the groundwater conditions, values of Poisson s ratio and different sets of shear strength parameters, a total of 48 cases were analyzed. A summary of the soil properties used in this case study is presented in Table 1. The slope stability results obtained when using DYNPROG were compared with those produced by several limit equilibrium methods of slices, including Bishop s simplified method (Bishop 1955), the Morgenstern Price method (Morgenstern and Price 1965), and the Enhanced method (Fredlund and Scoular 1999). The finite element based interslice force function of Fan et al. (1986) was used in conjunction with the Morgenstern Price method. All limit equilibrium methods of slices were performed using SLOPE/W (Geo-Slope International Ltd. 2001c) and then compared with the DYNPROG results. The stress and seepage analyses associated with the Enhanced method were performed using SIGMA/W (Geo-Slope International Ltd. 2001b) and SEEP/W (Geo-Slope International Ltd. 2001a), respectively Stability analysis of the wet slope The wet condition for the simple slope is presented first. The factors of safety computed using the dynamic programming method were the lowest when Poisson s ratio was

7 836 Can. Geotech. J. Vol. 40, 2003 Table 1. Soil properties for the homogeneous slope. c (kpa) φ ( ) φ b ( ) Note: Two pore-water pressure conditions, wet slope and submerged slope, and two values of Poisson s ratio, µ = 0.33 and 0.48, were considered. equal to 0.33 (Fig. 6). The factors of safety calculated when using the limit equilibrium methods of slices (i.e., Morgenstern Price and Bishop s Simplified methods) were higher. The largest difference in values for the factor of safety calculated by various methods was observed when the angle of internal friction, φ, was 10 and the cohesion, c, was 20 kpa. The difference was as large as 15% between the factors of safety computed from DYNPROG and the Morgenstern Price method, with the factor of safety computed by DYNPROG being lower. The factors of safety calculated by the limit equilibrium methods were also found to be significantly different when the cohesion of the soil was high (i.e., c = 40 kpa). For the larger value of Poisson s ratio (i.e., µ = 0.48), it was observed that the factor of safety from DYNPROG consistently increased with an increase in Poisson s ratio. Furthermore, the factors of safety determined using DYNPROG became slightly higher than those obtained by the limit equilibrium methods (Fig. 7). The increase in the value of the factor of safety as Poisson s ratio increases has also been observed by Martins et al. (1981), Matos (1982), and Scoular (1997). The factors of safety calculated by conventional limit equilibrium methods of slices (i.e., the Morgenstern Price and Bishop s Simplified methods) are unaffected by Poisson s ratio, since the stresses acting at the base of a slice do not represent realistic stress fields within the slope (Krahn 2003). The comparison of the factors of safety computed by DYNPROG and the Morgenstern Price method are presented in Fig. 8. When Poisson s ratio was equal to 0.33, the location of the critical slip surface determined using DYNPROG (Fig. 9) was about midway between the locations of the critical slip surfaces found by the Morgenstern Price and Enhanced methods. For the greater value of Poisson s ratio (i.e., µ = 0.48), the critical slip surface determined by DYNPROG generally tended to go slightly deeper than the critical slip surfaces found by the limit equilibrium methods. Moreover, the shape of the critical slip surface changed from being circular towards a logarithmic spiral shape. The shape of the critical slip surface tended to be more circular in shape when Poisson s ratio increased from 0.33 to The calculation of the local factors of safety along the critical slip surface also depended on the value of Poisson s ratio, as shown in Fig. 10. For lower values of Poisson s ratio, the local factor of safety was high at the entry and exit points of the critical slip surface. Along the interior of the critical slip surface, the local safety factors remained relatively constant. When Poisson s ratio increased, the distribution of local factors of safety became fairly constant. Fluctuations in the distribution of local factors of safety appeared when the angle of internal friction was relatively high (i.e., φ = 30 ) Stability analysis of the submerged slope For the submerged condition, the factors of safety computed using the dynamic programming method were also lowest when Poisson s ratio was equal to 0.33 (Fig. 11). A similar comparison of factors of safety for the case when Poisson s ratio was 0.48 is shown in Fig. 12. The overall factors of safety determined in the case of a submerged slope were greater than those obtained in the case of the wet slope for the range of soil properties and Poisson s ratios selected for the study. This is as anticipated, since the slope is more stable when supported by water at its toe. Again, Poisson s ratio was recognized as having an effect on the stability of the slope in terms of the shape and the location of the critical slip surface and the values of the associated factors of safety. When Poisson s ratio was 0.33, the factors of safety computed for the submerged slope when using DYNPROG are lower than those calculated by the Enhanced and Morgenstern Price methods (Fig. 13). This difference was as large as 10% between DYNPROG and the Morgenstern Price method, as observed in the case where the cohesion was equal to 30 kpa and the angle of internal friction was 10, with the factor of safety computed by DYNPROG being lower. The critical slip surface determined by DYNPROG was also found to be midway between those found by the other methods. When Poisson s ratio was 0.48, the factors of safety determined by DYNPROG were again slightly higher those determined by the limit equilibrium methods. The largest difference in factors of safety computed by DYNPROG and the Morgenstern Price method was about 1.6%, which is not significant in engineering practice. Although there was no considerable difference in the factors of safety computed by the examined methods, the locations of the critical slip surfaces determined by these methods were distinguishable. The critical slip surfaces found when using DYNPROG were deeper than those found by the limit equilibrium methods (Fig. 14). Furthermore, the shape of the critical slip surface found by DYNPROG was smoother and more circular with a high Poisson s ratio than when Poisson s ratio was lower (i.e., µ = 0.33). The distribution of the local factors of safety along the critical slip surface was similar to that of the wet condition (Fig. 15). 3.2 Nonhomogeneous slope Three example problems of a nonhomogeneous slope were analyzed. Soils were assumed to behave in a linear elastic manner. Soil properties pertaining to each example

8 Pham and Fredlund 837 Fig. 6. Factor of safety versus Janbu s (1954) stability number for the wet condition (µ = 0.33). M P, Morgenstern Price method. Fig. 7. Factor of safety versus Janbu s stability number for the wet condition (µ = 0.48).

9 838 Can. Geotech. J. Vol. 40, 2003 Fig. 8. DYNPROG and Morgenstern Price factors of safety (F s ) for the wet condition. Fig. 9. Locations of the critical slip surfaces obtained by various methods for the wet condition. are listed in Table 2, and the locations of the critical slip surfaces are presented in Figs It was observed that the factors of safety computed by DYNPROG were slightly lower than those computed by other methods in all examples. There were also some variations in the location of the critical slip surfaces found by the methods examined. The slope analyzed in Fig. 16 comprised two soil layers that were not distinctly different. The critical slip surface

10 Pham and Fredlund 839 Fig. 10. The distribution of local factors of safety for the wet condition. Fig. 11. Factor of safety versus Janbu s stability number for the submerged condition (µ = 0.33). computed by the dynamic programming method showed a slightly deeper slip surface near the toe of the slope and the computed factor of safety was slightly lower than that obtained from the Morgenstern Price and Enhanced methods. The slope analyzed in Fig. 17 comprised two soil layers resting on a relatively hard base. Once again, the factors of safety computed using the dynamic programming method were similar to those computed by other methods. The location of the critical slip surface was essentially circular and compared well with other critical slip surfaces. The slope analyzed in Fig. 18 comprised three soil layers, with the thin middle layer consisting of relatively weak soil. The weak layer was used to force the shape of the critical slip surface into a composite mode. The factor of safety computed using the dynamic programming method was about 14% lower than that obtained from the Morgenstern

11 840 Can. Geotech. J. Vol. 40, 2003 Fig. 12. Factor of safety versus Janbu s stability number for the submerged condition (µ = 0.48). Fig. 13. DYNPROG and Morgenstern Price factors of safety for the submerged condition.

12 Pham and Fredlund 841 Fig. 14. Locations of the critical slip surfaces obtained by various methods for the submerged condition. Fig. 15. The distribution of local factors of safety for the submerged condition. Price method. The critical slip surface has a pronounced nonlinear shape enclosing the soil mass. This example problem illustrates the ease with which the dynamic programming method can locate an irregular critical slip surface of any shape The Lodalen case history The landslide in Lodalen, Oslo, Norway, has long been recognized as a thorough case history in the study of slope stability. The slide was well-documented and studied by Sevaldson (1956). The properties of the soil at the Lodalen site are summarized in Table 3. Although Poisson s ratio of the soil was not reported in the literature, its value can be somewhat anticipated from other data. The soil at the Lodalen site was stated to be slightly overconsolidated (Sevaldson 1956), and the lateral coefficient of the soil pressure at rest, K 0, should be slightly greater than that of a normally consolidated soil. From the

13 842 Can. Geotech. J. Vol. 40, 2003 Table 2. Soil properties of three examples of a nonhomogeneous slope. Layer Unit weight (kn/m 3 ) Shear strength parameters Poisson s Young s ratio c (kpa) φ φ b modulus (kpa) Example 1 Upper Lower Example 2 Upper Middle Lower Example 3 Medium Weak Hard Fig. 16. Nonhomogeneous slope with two soil layers. average value for the angle of internal friction presented in Table 3, the value of K 0 for a normally consolidated soil can be calculated as follows (Jaky 1944): [15] K 0 =1 sinφ where φ is the effective friction angle. Poisson s ratio, µ, is related to K 0 by the following relationship: K [16] µ= K 0 Using the average value of the angle of internal friction of the soil in Table 3, Poisson s ratio of the soil at the Lodalen site should be greater than Since the location of the critical slip surface depends on the value of Poisson s ratio, it is of interest to vary the value of Poisson s ratio to observe the change in the location of the critical slip surface. It was, therefore, assumed that Poisson s ratio ranged from 0.37 to 0.42 in this analytical study. The reanalysis of slide 2 of the Lodalen case history, utilizing several values of Poisson s ratio, yielded a zone of critical slip surfaces. As shown in Fig. 19, the location of the actual slip surface ranged between lower and the upper limits corresponding to the selected values of Poisson s ratio. It is apparent that there must be a value of Poisson s ratio that best corresponds to the location of the actual slip surface. The most likely value for Poisson s ratio also corresponds well within the location of the critical slip surface.

14 Pham and Fredlund 843 Fig. 17. Nonhomogeneous slope resting on a hard base. Fig. 18. Nonhomogeneous slope with an extremely weak layer. 4. Sensitivity studies The solution of the slope stability analysis using the dynamic programming method depends on several factors, such as finite element stress distributions and geometrical conditions imposed on the search grid. Different stress distributions obtained from different approaches in the finite element stress analysis may have an effect on the shape and

15 844 Can. Geotech. J. Vol. 40, 2003 Fig. 19. The Lodalen case history, slide 2. Table 3. Soil properties at the Lodalen slide 2 (after Sevaldson 1956). Boring Depth (m) Effective cohesion (kpa) Average Effective angle of friction ( ) location of the critical slip surface and the overall factor of safety. Moreover, the density of the search grid may also influence the smoothness of the slip surface obtained and the computing time required to solve the problem. 4.1 Influence of different stress distributions Clough and Woodward (1967) stated that the use of a nonlinear stress analysis was essential if the displacements or deformations were of main interest. Otherwise, the stress distribution can be reasonably determined using either a linear or a nonlinear stress analysis. Slope stability results obtained from linear and nonlinear stress analyses are compared in this section. Stresses within a wet homogeneous slope at 2:1, which is similar to the wet slope presented in Sect , were analyzed using both linear and nonlinear elastic models. In the linear elastic stress analysis, the stress distribution was obtained by simply switching on the gravity of the soil within the boundaries of the soil mass. A hyperbolic constitutive model (Duncan and Chang 1970) was adopted in the nonlinear stress analysis. In this approach, the slope was assumed to be constructed in 10 successive lifts from the base. The tangential modulus of the soil was updated based on the previous stress state when a next lift was placed. The locations of the critical slip surfaces determined by DYNPROG using stresses from both linear elastic and nonlinear elastic stress analyses are presented in Fig. 20. The critical slip surface determined when using a nonlinear elastic stress analysis was slightly deeper than that obtained when using a linear elastic stress analysis. The coincidence of locations of critical slip surfaces illustrates that there is little difference in the distribution of stresses when using a linear elastic and a nonlinear elastic stress analysis. The difference in the factors of safety calculated using both approaches was also not of significance. The observed differences were less than approximately 4.5% between the linear and nonlinear elastic stress analysis. The nonlinear analysis gave higher values for the factor of safety. 4.2 Sensitivity of the search grid density The optimization search for the dynamic programming method was performed on a grid of stage state points. As previously described, this grid is referred to as the search grid. In this section, there are four densities of the search grid examined, including the coarse (5 m 1 m) grid, the medium (2 m 0.5 m) grid, the fine (2 m 0.25 m) grid,

16 Pham and Fredlund 845 Fig. 20. Stability analyses based on different stress distributions. and the dense (1 m 0.25 m) grid. The common density of the search grid used in previous sections of this study was 2 m 0.25 m, which corresponds to the fine grid. In general, the denser search grid gave a computed critical slip surface with a smoother shape. The computing time required to solve a given problem increases significantly, however, with an increase in the density of the search grid. As shown in Fig. 21, locations of the critical slip surfaces obtained for all densities were in fairly good agreement, with the exception of the coarse grid. The critical slip surface produced by the coarse grid was rough and the location of the entry point of the critical slip surface was distinctly different as compared with those obtained when using other densities. The roughness of critical slip surfaces determined using the remaining grids was essentially the same. It was also noted that factors of safety calculated from different search grid densities were quite insensitive to the densities of the search grids. The computing time required to solve the problem significantly depended on the density of the search grid. As shown in Fig. 21, the computing time increased from 2 s per iteration for the coarse grid to 325 s per iteration for the dense grid. Moreover, the computing time was quite sensitive to the number of state points used. The difference between the amount of computing time for the fine grid and that for the medium grid was as large as 6.5 times, whereas the difference between the fine grid and the dense grid almost doubled. Baker (1980) suggested that the ratio of the distance between two successive state points over the distance between two successive stage points should be about one to four, which corresponds to the medium grid in this study. The fine grid, which was used in all analyses presented in previous sections, had a ratio of one to eight. It can be concluded that the density of the search grid does not seriously affect the results of the analysis in terms of the value of the factor of safety. The medium grid appears to be satisfactory when considering both the factor of safety results and the required computing time. A coarse grid can also be used for the initial run to anticipate the potential location of the critical slip surface and the initial value of the factor of safety. The final search grid can be further densified with a focus on the zone of the critical slip surface obtained from the first run. Theoretical significance of the findings The findings of the comparative study between the dynamic programming solutions using DYNPROG and the other methods of slope stability analysis appear to be consistent with what would be expected from a theoretical standpoint. For example, both the dynamic programming solution and the Enhanced method yield critical slip surfaces that go slightly below the toe of the slope, whereas critical slip surfaces found by limit equilibrium methods of analysis appeared to exit higher and nearer to the toe of the slope for the simple slopes analyzed. This is as anticipated, since the limit equilibrium methods of slices compute the normal force on the base of a slice without consideration of the ground surface geometry at adjacent slices. On the other hand, the stresses computed from a finite element analysis take the ground surface geometry into consideration and

17 846 Can. Geotech. J. Vol. 40, 2003 Fig. 21. Sensitivity of various densities of the search grid. therefore stress concentrations at the toe of the slope are taken into account. All of the limit equilibrium methods of slices tend towards an upper bound type of solution because the shape of the slip surface is controlled by the analyst (Ching and Fredlund 1984). Consequently, it is anticipated that the computed factors of safety would tend towards being slightly higher than the correct solution. Various limit equilibrium methods of analysis can yield slightly different factors of safety as a consequence of the assumption invoked to render the analysis determinate, but all results will tend towards an upper bound solution from a plasticity standpoint. On the other hand, the shape of the slip surface is not dictated by the analyst in the dynamic programming method. Consequently, the solution tends towards a more correct solution, and the computed factors of safety should be slightly lower. This behaviour was consistently observed in the comparative study undertaken when Poisson s ratio was Although the factors of safety computed from the dynamic programming technique are lower than (or at least equal to) those computed by limit equilibrium analyses, the difference is relatively small for the simple slope examples studied. The small difference is encouraging because the increased flexibility of the dynamic programming technique opens the way for new possibilities for analyzing complex slope stability problems. When a Poisson s ratio of 0.48 was used in the stress analysis, the computed factors of safety increase and are similar to (or even greater than) those computed from the Morgenstern Price method. It is possible that a Poisson s ratio approaching 0.5 corresponds to zero volume change (or rigid body motion), and for this reason the computed factors of safety are similar to those obtained from the limit equilibrium solution. Further study would be of benefit regarding this point. The shape of the critical slip surface deviates somewhat from a circular shape, even for a homogeneous soil slope, for the dynamic programming solution. Consequently, it would be anticipated that the computed factor of safety would be slightly lower than that obtained from the Enhanced method where the shape of the critical slip surface is controlled (i.e., circular in this case). In other words, the dynamic programming method has been able to locate a slip surface shape that is slightly more efficient (in terms of minimizing the factor of safety function) than a circular shape, even for a homogeneous soil slope. All of the findings from this study tend to point towards a movement away from a complete reliance upon limit equilibrium methods of slope stability analyses. The difference between the dynamic programming technique and the limit equilibrium technique is relatively small for simple slopes, but the flexibility of the dynamic programming technique opens the door for the analysis of problems that can be better understood in conjunction with a stress analysis. 6. Conclusions The dynamic programming method combined with a finite element stress analysis can be a viable and valuable tool for practical slope stability analyses. With the use of the finite element stress analysis, the present method provides a solution of greater flexibility compared with those produced by conventional limit equilibrium methods of slices. Another advancement of the dynamic programming method, compared to the Enhanced method, is that the critical slip surface can be irregular in shape and, more impor-

18 Pham and Fredlund 847 tantly, can be determined as part of the slope stability solution. Previously, the mode of failure needed to be anticipated by an analyst, but now the mode of failure is part of the solution. In other words, there is no assumption required regarding the shape or the location of the critical slip surface except the assumption that the critical slip surface is an assemblage of linear segments. More complex and rigorous stress strain behaviours of the soil such as nonlinear, elastoplastic models can be used in the finite element stress analysis. Therefore, the effects of the stress history and volume-change behaviour during shear can be taken into consideration as part of the analysis. The effect of weather-related environmental conditions such as infiltration (or matric suction decrease) on the stability of a slope can also be studied when the dynamic programming method is coupled with a transient, finite element seepage analysis. Acknowledgement The financial support of the Canadian International Development Agency (CIDA) in conducting this research is highly acknowledged. The authors believe that collaboration and technology transfer can prove to be of considerable benefit to both developing and developed countries. References Baker, R Determination of the critical slip surface in slope stability computations. International Journal for Numerical and Analytical Methods in Geomechanics, 4: Bellman, R Dynamic programming. Princeton University Press, Princeton, N.J. Bishop, A.W The use of the slip circle in the stability analysis of slopes. Géotechnique, 5: Celestino, T.B., and Duncan, J.M Simplified search for noncircular slip surfaces. In Proceedings of the 10th International Conference on Soil Mechanics and Foundation Engineering, Stockholm, June Vol. 3. A.A. Balkema, Rotterdam, The Netherlands. pp Chen, Z.-Y., and Shao, C.-M Evaluation of minimum factor of safety in slope stability analysis. 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